;; The first four lines of this file were added by Dracula.
;; They tell DrScheme that this is a Dracula Modular ACL2 program.
;; Leave these lines unchanged so that DrScheme can properly load this file.
#reader(planet "reader.rkt" ("cce" "dracula.plt") "modular" "lang")
;;=======  Module: AVL trees   ================================================
;; Defining
;;   (avl-retrieve tr key)
;;   (empty-tree)
;;   (avl-insert tr key datum)
;;   (avl-delete tr key)
;;   (avl-flatten tr)
;;=============================================================================
;; Module usage notes
;;
;; Place this file in the following directory
;;    C:/ACL2-3.1/acl2-sources/books/SE/
;;
;;  Use the following command to certify the module
;;  (certify-book "C:/ACL2-3.1/acl2-sources/books/SE/avl-rational-keys")
;;
;; Command to get access to module functions (any session or program)
;;  (include-book "SE/avl-rational-keys" :dir :system)
;;
;;=============================================================================
;; Function usage notes
;;
;; 1. (avl-retrieve tr key)
;;   assumes
;;     tr         has been constructed by one of the AVL-tree constructors
;;                (empty-tree, avl-insert, and avl-delete)
;;     new-key    is a rational number
;; 
;;   delivers
;;     either a two element list (k d)
;;       such that k equals key and
;;                 tr contains a subtree with k and d in its root
;;     or nil, in case key does not occur in tr
;;   
;; 2. (empty-tree)
;;   delivers an empty AVL-tree
;;
;; 3. (avl-insert tr key datum)
;;   assumes
;;      tr       has been constructed by the AVL-tree constructors
;;               (empty-tree, avl-insert, or avl-delete)
;;      key      is a rational number
;;   delivers an AVL tree with the following property
;;     (and (equal (avl-retrieve (avl-insert tr key datum) key)
;;                 (list key datum))
;;          (iff (avl-retrieve (avl-insert tr key datum) k)
;;               (or (avl-retrieve tr k)
;;                   (key= k key))))
;;
;; 4. (avl-delete tr key)
;;   assumes
;;      tr       has been constructed by the AVL-tree constructors
;;               (empty-tree, avl-insert, and avl-delete)
;;      key      is a rational number
;;   delivers an AVL tree with the following property
;;     (equal (avl-retrieve (avl-delete tr key) key)
;;            nil)
;;
;; 5. (avl-flatten tr)
;;   assumes
;;      tr       has been constructed by the AVL-tree constructors
;;               (empty-tree, avl-insert, and avl-delete)
;;   delivers a list of cons-pairs with the following properties
;;      (and (implies (occurs-in-tree? k tr)
;;                    (and (occurs-in-pairs? k (avl-flatten tr))
;;                         (meta-property-DF tr k)))
;;           (implies (not (occurs-in-tree? k tr))
;;                    (not (occurs-in-pairs? k (avl-flatten tr)))))
;;           (increasing-pairs? (avl-flatten tr)))
;;     where (meta-property-DF tr k) means that one of the elements, e, in
;;     the list (avl-flatten tr) satisfies (equal (car e) k)) and
;;     (cadr e) is the datum at the root of the subtree of tr where k occurs
;;
;;=============================================================================
;;Environment setup
;  (in-package "ACL2")
;;=============================================================================

(require "I-avl-string-keys.lisp")

(module Mtree
  ; Extractors (and empty-tree detector)
  (defun empty-tree? (tr) (not (consp tr)))
  (defun height (tr) (if (empty-tree? tr) 0 (car tr)))
  (defun key (tr) (cadr tr))
  (defun data (tr) (caddr tr))
  (defun left (tr) (cadddr tr))
  (defun right (tr) (car (cddddr tr)))
  (defun keys (tr)
    (if (empty-tree? tr)
        nil
        (append (keys (left tr)) (list (key tr)) (keys (right tr)))))
  
  ; Constructors
  (defun empty-tree ( ) nil)
  (defun tree (k d lf rt)
    (list (+ 1 (max (height lf) (height rt))) k d lf rt))
  
  ; Contstraint detectors and key comparators
  (defun key? (k) (stringp k))	  ; to change representation of keys
  (defun key< (j k) (string< j k))	  ;     alter definitions of key? and key<
  (defun key> (j k) (key< k j))
  (defun key= (j k)		  ; note: definitions of
    (and (not (key< j k))           ;    key>, key=, and key-member	
         (not (key> j k))))	  ;        get in line automatically
  (defun key-member (k ks)
    (and (consp ks)
         (or (key= k (car ks))
             (key-member k (cdr ks)))))
  (defun data? (d)
    (if d t t))
  (defun tree? (tr)
    (or (empty-tree? tr)
        (and (natp (height tr))		       ; height
             (= (height tr)                      ;   constraints
                (+ 1 (max (height (left tr))
                          (height (right tr)))))
             (key? (key tr))                     ; key constraint
             (data? (data tr))                   ; data constraint
             (tree? (left tr))                   ; subtree
             (tree? (right tr)))))               ;   constraints
  
  ; Key occurs in tree detector
  (defun occurs-in-tree? (k tr)
    (and (key? k)
         (tree? tr)
         (key-member k (keys tr))))
  (defun alternate-occurs-in-tree? (k tr)
    (and (key? k)
         (tree? tr)
         (not (empty-tree? tr))
         (or (key= k (key tr))
             (alternate-occurs-in-tree? k (left tr))
             (alternate-occurs-in-tree? k (right tr)))))
  
  ; all-key comparators
  (defun all-keys< (k ks)
    (or (not (consp ks))
        (and (key< (car ks) k) (all-keys< k (cdr ks)))))
  
  (defun all-keys> (k ks)
    (or (not (consp ks))
        (and (key> (car ks) k) (all-keys> k (cdr ks)))))
  
  ; definitions of ordered and balanced, and avl-tree detector
  (defun ordered? (tr)
    (or (empty-tree? tr)
        (and (tree? tr)
             (all-keys< (key tr) (keys (left tr)))
             (all-keys> (key tr) (keys (right tr)))
             (ordered? (left tr))
             (ordered? (right tr)))))
  
  (defun balanced? (tr)
    (and (tree? tr)
         (or (empty-tree? tr)
             (and (<= (abs (- (height (left tr)) (height (right tr)))) 1)
                  (balanced? (left tr))
                  (balanced? (right tr))))))
  
  (defun avl-tree? (tr)
    (and (ordered? tr)
         (balanced? tr)))
  
  ; rotations
  (defun easy-R (tr)
    (let* ((z (key tr)) (dz (data tr))
                        (zL (left tr)) (zR (right tr))
                        (x (key zL)) (dx (data zL))
                        (xL (left zL)) (xR (right zL)))
      (tree x dx xL (tree z dz xR zR))))
  
  (defun easy-L (tr)
    (let* ((z (key tr)) (dz (data tr))
                        (zL (left tr)) (zR (right tr))
                        (x (key zR)) (dx (data zR))
                        (xL (left zR)) (xR (right zR)))
      (tree x dx (tree z dz zL xL) xR)))
  
  (defun left-heavy? (tr)
    (and (tree? tr)
         (not (empty-tree? tr))
         (= (height (left tr)) (+ 2 (height (right tr))))))
  
  (defun outside-left-heavy? (tr)
    (and (left-heavy? tr)
         (or (= (height (left (left tr)))
                (height (right (left tr))))
             (= (height (left (left tr)))
                (+ 1 (height (right (left tr))))))))
  
  (defun right-rotatable? (tr)
    (and (ordered? tr)
         (not (empty-tree? tr))
         (balanced? (left tr))
         (balanced? (right tr))
         (not (empty-tree? (left tr)))))
  
  (defun right-heavy? (tr)
    (and (tree? tr)
         (not (empty-tree? tr))
         (= (height (right tr)) (+ 2 (height (left tr))))))
  
  (defun outside-right-heavy? (tr)
    (and (right-heavy? tr)
         (or (= (height (right (right tr))) (height (left (right tr))))
             (= (height (right (right tr))) (+ 1 (height (left (right tr))))))))
  
  (defun left-rotatable? (tr)
    (and (tree? tr)
         (not (empty-tree? tr))
         (balanced? (left tr))
         (balanced? (right tr))
         (not (empty-tree? (right tr)))))
  
  (defun hard-R (tr)
    (let* ((z (key tr))
           (dz (data tr))
           (zL (left tr))
           (zR (right tr)))
      (easy-R (tree z dz (easy-L zL) zR))))
  
  (defun hard-L (tr)
    (let* ((z (key tr))
           (dz (data tr))
           (zL (left tr))
           (zR (right tr)))
      (easy-L (tree z dz zL (easy-R zR)))))
  
  (defun inside-left-heavy? (tr)
    (and (left-heavy? tr)
         (= (height (right (left tr)))
            (+ 1 (height (left (left tr)))))))
  
  (defun hard-R-rotatable? (tr)
    (and (right-rotatable? tr)
         (left-rotatable? (left tr))))
  
  (defun inside-right-heavy? (tr)
    (and (right-heavy? tr)
         (= (height (left (right tr)))
            (+ 1 (height (right (right tr)))))))
  
  (defun hard-L-rotatable? (tr)
    (and (left-rotatable? tr)
         (right-rotatable? (right tr))))
  
  (defun rot-R (tr)
    (let ((zL (left tr)))
      (if (< (height (left zL)) (height (right zL)))
          (hard-R tr)
          (easy-R tr))))
  
  (defun rot-L (tr)
    (let ((zR (right tr)))
      (if (< (height (right zR)) (height (left zR)))
          (hard-L tr)
          (easy-L tr))))
  
  ; insertion
  (defun avl-insert (tr new-key new-datum)
    (if (empty-tree? tr)
        (tree new-key new-datum (empty-tree) (empty-tree))
        (if (key< new-key (key tr))
            (let* ((subL (avl-insert (left tr) new-key new-datum))
                   (subR (right tr))
                   (new-tr (tree (key tr) (data tr) subL subR)))
              (if (= (height subL) (+ (height subR) 2))
                  (rot-R new-tr)
                  new-tr))
            (if (key> new-key (key tr))
                (let* ((subL (left tr))
                       (subR (avl-insert (right tr) new-key new-datum))
                       (new-tr (tree (key tr) (data tr) subL subR)))
                  (if (= (height subR) (+ (height subL) 2))
                      (rot-L new-tr)
                      new-tr))
                (tree new-key new-datum (left tr) (right tr))))))
  
  ; delete root - easy case
  (defun easy-delete (tr)
    (right tr))
  
  ; tree shrinking
  (defun shrink (tr)
    (if (empty-tree? (right tr))
        (list (key tr) (data tr) (left tr))
        (let* ((key-data-tree (shrink (right tr)))
               (k (car key-data-tree))
               (d (cadr key-data-tree))
               (subL (left tr))
               (subR (caddr key-data-tree))
               (shrunken-tr (tree (key tr) (data tr) subL subR)))
          (if (= (height subL) (+ 2 (height subR)))
              (list k d (rot-R shrunken-tr))
              (list k d shrunken-tr)))))
  
  (defun raise-sacrum (tr)
    (let* ((key-data-tree (shrink (left tr)))
           (k (car key-data-tree))
           (d (cadr key-data-tree))
           (subL (caddr key-data-tree))
           (subR (right tr))
           (new-tr (tree k d subL subR)))
      (if (= (height subR) (+ 2 (height subL)))
          (rot-L new-tr)
          new-tr)))
  
  ; delete root - hard case
  (defun delete-root (tr)
    (if (empty-tree? (left tr))
        (easy-delete tr)
        (raise-sacrum tr)))
  
  ; deletion
  (defun avl-delete (tr k)
    (if (empty-tree? tr)
        tr
        (if (key< k (key tr))           ; key occurs in left subtree
            (let* ((new-left (avl-delete (left tr) k))
                   (new-tr (tree (key tr) (data tr) new-left (right tr))))
              (if (= (height (right new-tr)) (+ 2 (height (left new-tr))))
                  (rot-L new-tr)
                  new-tr))
            (if (key> k (key tr))       ; key occurs in right subtree
                (let* ((new-right (avl-delete (right tr) k))
                       (new-tr (tree (key tr) (data tr) (left tr) new-right)))
                  (if (= (height (left new-tr)) (+ 2 (height (right new-tr))))
                      (rot-R new-tr)
                      new-tr))
                (delete-root tr)))))  ; key occurs at root
  
  ; retrieval
  (defun avl-retrieve (tr k)  ; delivers key/data pair with key = k
    (if (empty-tree? tr)      ; or nil if k does not occur in tr
        nil                                 ; signal k not present in tree
        (if (key< k (key tr))
            (avl-retrieve (left tr) k)      ; search left subtree
            (if (key> k (key tr))
                (avl-retrieve (right tr) k) ; search right subtree
                (cons k (data tr))))))      ; k is at root, deliver key/data pair
  
  (defun avl-flatten (tr)  ; delivers all key/data cons-pairs
    (if (empty-tree? tr)   ; with keys in increasing order
        nil
        (append (avl-flatten (left tr))
                (list (cons (key tr) (data tr)))
                (avl-flatten (right tr)))))
  
  (defun occurs-in-pairs? (k pairs)
    (and (consp pairs)
         (or (key= k (caar pairs))
             (occurs-in-pairs? k (cdr pairs)))))
  
  (defun increasing-pairs? (pairs)
    (or (not (consp (cdr pairs)))
        (and (key< (caar pairs) (caadr pairs))
             (increasing-pairs? (cdr pairs)))))
  (export Itree))